A Splitting Method for Nonlinear Filtering Problems with Diffusive and Point Process Observations

This paper aims to develop and analyze a comprehensive discretized splitting-up numerical scheme for the Zakai equation. This equation arises from a nonlinear filtering problem, where observations incorporate noise modeled by point processes and Wiener processes. Initially, we introduce a regularization parameter and employ a splitting-up approach to break down the Zakai equation into two stochastic differential equations and a partial differential equation (PDE). Subsequently, we employ a finite difference scheme for the temporal dimension and the spectral Galerkin method for the spatial dimension to achieve full discretization of these equations. This results in a numerical solution for the Zakai equation using the splitting-up technique. We demonstrate that this numerical solution converges to the exact solution with a convergence order of 12. Additionally, we conduct several numerical experiments to illustrate and validate our theoretical findings.